Geometry and Topology of Manifolds with Special Holonomy

特殊完整流形的几何与拓扑

• 批准号: 1105663
• 负责人: Sema Salur
• 金额: $ 12.67万
• 依托单位: University of Rochester
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-08-15 至 2016-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1105663&HistoricalAwards=false
• 关键词: Geometry; Topology; Manifolds; Special; Holonomy

Manifolds with special holonomy and calibrated geometry have important relations with symplectic geometry, complex geometry, algebraic geometry, Cartan-Kahler Theory and Seiberg-Witten Theory. In this project, the P.I. plans to continue her work on calibrated submanifolds of Calabi-Yau, G_2 and Spin(7) manifolds. In joint work with S. Akbulut, the P.I. studied complex associative and complex Cayley submanifolds and using the SW theory she showed that the moduli space of these submanifolds is smooth without any obstructions to such deformations. She also introduced the mathematical definitions of mirror Calabi-Yau and G_2 manifolds. The P.I. plans to continue her work on compactification problems of these moduli spaces and to study the mirror dualities in other Calabi-Yau and G_2 manifolds. These will lead to the construction of new counting invariants and provide a better understanding of the mirror symmetry phenomenon. The research topics of this project are motivated by questions in geometry and topology of low dimensional manifolds and mathematical physics. The P.I. believes that this makes calibrated geometry an excellent subject for students who might decide to study math and science, and as a female researcher in this exciting area she feels a particular responsibility to encourage young women to begin and to continue their studies of mathematics. She will continue to encourage both undergraduate and graduate students and to collaborate with them in this research field which is expected to have a long-lasting impact on both mathematics and physics.

具有特殊的自动化和校准几何形状的歧管与符号几何形状，复杂几何形状，代数几何，cartan-kahler理论和塞伯格 - 理论具有重要关系。在这个项目中，P.I.计划继续在校准的Calabi-Yau，G_2和Spin（7）歧管的校准子手法上工作。在与P.I. S. akbulut的联合合作中研究了复杂的关联和复杂的Cayley Submanifolds，并使用SW理论表明，这些亚策略的模量空间是光滑的，没有任何这种变形的障碍。她还介绍了镜子calabi-yau和g_2歧管的数学定义。 P.I.计划继续她在这些模量空间的压缩问题上的工作，并研究其他卡拉比Yau和G_2歧管中的镜子二元性。这些将导致构建新的计数不变，并更好地理解镜像对称现象。该项目的研究主题是由低维流形和数学物理学的几何学和拓扑中的问题激励的。 P.I.认为这使得经过校准的几何形状成为可能决定学习数学和科学的学生，并且作为这个令人兴奋的领域的女性研究员，她感到特别责任鼓励年轻女性开始并继续对数学研究进行研究。她将继续鼓励本科生和研究生，并在这个研究领域与他们合作，这将对数学和物理学产生持久的影响。